20 2. MATHEMATICAL CULTURES Iarithmetic and geometry, they come to astronomy, and Socrates (the narrator of
the dialogue) reports that Glaucon was in favor of including it:"I certainly am," he said, "to have a clearer perception of time peri-
ods, both months and eras is proper in agriculture and navigation,
and no less so in military strategy."
"You amuse me," I said, "in that you apparently fear the
crowd, lest you seem to prescribe useless studies. But it is not
an easy thing, it is difficult, to believe that some organ of the soul
of each person is purified and refreshed in these studies, while it is
lost and blinded by other pursuits; it is more to be preserved than
ten thousand eyes, for by it alone is truth seen. Now some peo-
ple will agree that what you are proposing is extremely good; but
those who have never felt these things will regard you as having
said nothing; for they see in them no profit worthy of the name."Plato adopted the Pythagorean doctrine that there is a human faculty attuned
to eternal truth and working through human reason. If there really is such a
faculty, then of course mathematics is of high value for everyone. Plato, speaking
here through Socrates, admits that some people seem to lack this faculty, so that
it is difficult to believe in its universality. The difference in outlook between the
two classes of people mentioned by Plato continues right down to the present time.
Here, for example, is the view of the famous modern applied mathematician, R. W.
Hamming (1915-1998), inventor of the Hamming codes, who if he believed at all in
the "eye of the soul," at least did not believe it had a claim on public funds:
[T]he computing expert needs to be wary of believing much that
he learns in his mathematics courses... [M]uch of modern math-
ematics is not related to science but... to the famous scholastic
arguing of the Middle Ages... I believe it is important to make
these distinctions... [T]he failure to do so has. ..caused govern-
ment money appropriated for numerical analysis to be diverted to
the art form of pure mathematics.On the opposite side of the question is the following point of view, expressed
by the famous British mathematician G. H. Hardy (1877-1947), who in 1940 wrote
A Mathematician's Apology. After quoting an earlier address, in which he had said,
"after all, the scale of the universe is large and, if we are wasting our time, the
waste of the lives of a few university dons is no such overwhelming catastrophe,"^1
he gave what he considered the justification of his life:
I have never done anything "useful." No discovery of mine has
made, or is likely to make, directly or indirectly, for good or ill,
any difference to the amenity of the world... The case for my life,
then,..., is this: that I have added something to knowledge, and
have helped others to add more; and that these somethings have
a value which differs in degree only, not in kind, from that of the
creations of the great mathematicians.(^1) From a cosmic point of view, no doubt, this is a cogent argument. From the point of view of
that second group of people mentioned by Plato, however, it ignores the main question: Why
should a person expect to receive a salary for doing work that others regard as useless?